If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real, then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$

If $\theta=\frac{\pi}{6}$, then the 10 th term of the series $1+(\cos \theta+i \sin \theta)^1+(\cos \theta+i \sin \theta)^2+\ldots$. is

If $\alpha$ and $\beta$ are non-zero integers and $z=(\alpha+i \beta)(2+7 i)$ is a purely imaginary number, then minimum value of $|z|^2$ is